Risk analysis of dynamic systems: Markovian approach

The Markovian approach is the oldest, and therefore the best known and most widely used, of the methods implemented for the probabilistic treatment of dynamically behaving systems.

It falls into the category of "state-analytical approaches" based on identifying the different states of the system concerned, and then analyzing the evolution of the system between these states.

One of its most interesting features is that it can be represented graphically, which means it can be used without having to know the underlying mathematical details. However, as we must beware of "black-box" implementations, which are often a source of errors and misunderstandings, this dossier focuses on the essential principles of this mathematics, in relation to the concrete problems encountered by analysts.

In addition to its current use for classical reliability and availability calculations, this approach conceals resources that are often unsuspected even by its most assiduous users. That's why this dossier highlights the approach's ability to handle systems with degraded states, such as production systems ("multi-state" systems), for example, or/and several operating phases, such as periodically tested safety systems ("multi-phase" systems).

In addition to the usual evaluation of the probabilities of the various states of the system under study, this dossier shows how the evaluation of cumulative mean residence times (TMSC) spent in the various states opens the way to the treatment of a whole class of studies oriented towards economics rather than safety, and how, for example, the notion of average availability extends naturally into production availability.

Although very flexible and powerful, this approach suffers from a number of limitations, mainly stemming from the impossibility of using other than exponential laws, and from the combinatorial explosion in the number of states as the number of elementary components increases. This limits its rigorous implementation to small systems, but the approximate methods described in this dossier enable us to push back these limits to a certain extent.

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